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General Relativity and Quantum Cosmology

Title:
Averaged Energy Conditions and Quantum Inequalities

Abstract: Connections are uncovered between the averaged weak (AWEC) and averaged null
(ANEC) energy conditions, and quantum inequality restrictions on negative
energy for free massless scalar fields. In a two-dimensional compactified
Minkowski universe, we derive a covariant quantum inequality-type bound on the
difference of the expectation values of the energy density in an arbitrary
quantum state and in the Casimir vacuum state. From this bound, it is shown
that the difference of expectation values also obeys AWEC and ANEC-type
integral conditions. In contrast, it is well-known that the stress tensor in
the Casimir vacuum state alone satisfies neither quantum inequalities nor
averaged energy conditions. Such difference inequalities represent limits on
the degree of energy condition violation that is allowed over and above any
violation due to negative energy densities in a background vacuum state. In our
simple two-dimensional model, they provide physically interesting examples of
new constraints on negative energy which hold even when the usual AWEC, ANEC,
and quantum inequality restrictions fail. In the limit when the size of the
space is allowed to go to infinity, we derive quantum inequalities for timelike
and null geodesics which, in appropriate limits, reduce to AWEC and ANEC in
ordinary two-dimensional Minkowski spacetime. We also derive a quantum
inequality bound on the energy density seen by an inertial observer in
four-dimensional Minkowski spacetime. The bound implies that any inertial
observer in flat spacetime cannot see an arbitrarily large negative energy
density which lasts for an arbitrarily long period of time.